Sigmoid functions, Hypergeometric series, Legendre equation, Fourier transform
Sigmoid functions, whose graphs are "S-shaped" curves, appear in a great variety of contexts, such as the transfer functions in many neural networks. Their ubiquity is no accident; these curves are the among the simplest non-linear curves, striking a graceful balance between linear and non-linear behavior.
Menon, Anil Ravindran; Mehrotra, Kishan; Mohan, Chilukuri K.; and Ranka, Sanjay, "Characterization of a Class of Sigmoid Functions with Applications to Neural Networks" (1994). Electrical Engineering and Computer Science Technical Reports. 152.